NUMERICAL MODELING OF HEAT TRANSFER IN BIOLOGICAL TISSUE DOMAIN USING THE FUZZY FINITE DIFFERENCE METHOD

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1 6th Europan Confrnc on Computatonal Mchancs (ECCM 6) 7th Europan Confrnc on Computatonal Flud Dynamcs (ECFD 7) 5 Jun 08, Glasgow, UK NUMERICL MODELING OF HET TRNSFER IN BIOLOGICL TISSUE DOMIN USING THE FUZZY FINITE DIFFERENCE METHOD LICJ PISECK-BELKHYT¹, PWEŁ KOWLSKI² ¹Slsan Unvrsty of Tchnology Konarskgo 8, Glwc, Poland alcja.pascka@polsl.pl ²Embd Pawł Kowalsk Gdańska 48C/8, 4-89 Zabrz, Poland pawl.kowalsk@pl.s.u Ky words: Bo-hat Transfr, Fuzzy Fnt Dffrnc Mthod, Fuzzy Numbrs, Drctd Intrval rthmtc. bstract. In ths papr, th numrcal analyss of a hat transfr procss procdng n th non-homognous bologcal tssu doman s prsntd. on-dmnsonal problm s consdrd. ddtonally, n th mathmatcal modl, th thrmophyscal paramtrs of ach skn s layr (pdrms, drms, subcutanous rgon) such as volumtrc spcfc hat, thrmal conductvty, prfuson coffcnt and mtabolc hat sourc ar gvn as fuzzy numbrs. Th bas of mathmatcal modl s gvn by a st of Pnns fuzzy quatons supplmntd by th adquat boundary and ntal condtons. Th problm dscussd has bn solvd usng th fuzzy fnt dffrnc mthod wth α-cuts. Th xampls of numrcal computatons ar prsntd n th fnal part of th papr. INTRODUCTION Thrmophyscal paramtrs of bologcal tssu (thrmal conductvty, volumtrc spcfc hat, prfuson coffcnt tc.) can chang n a wd rang and ths s causd by ndvdual trats as ag, sx, occupaton tc. Ths fact suggsts th applcaton of fuzzy valus of ths paramtrs to mathmatcal modl of procss consdrd. Such an approach n whch th paramtrs apparng n th mathmatcal modl ar tratd as constant valus s wdly usd [,,3]. Hr, th fuzzy valus of thrmal conductvty, volumtrc spcfc hat, prfuson coffcnt and mtabolc hat sourc ar takn nto account. In th papr th bo-hat transfr procdng n a on-dmnsonal skn tssu dscrbd by th govrnng quatons s consdrd. Th problm has bn solvd usng a fuzzy vrson of th fnt dffrnc mthod (FDM) wth -cuts and th ruls of drctd ntrval arthmtc [4,5]. Th applcaton of -cuts allows on to avod vry complcatd arthmtcal opratons n th fuzzy numbrs st bcaus th -cuts ar closd ntrvals. Th man advantag of th drctd ntrval arthmtc upon th usual ntrval arthmtc s that th obtand tmpratur ntrvals ar much narrowr and thr wdth dos not ncras n tm [6,7].

2 . Pascka-Blkhayat, P. Kowalsk In th fnal part of th papr th xampls of numrcal smulatons ar prsntd for th skn tssu doman subjctd to xtrnal hat sourc. THE FUZZY NUMBERS Th ground of th mathmatcal ruls usd n ths papr s gvn by th fuzzy st thory. Ths approach s not common n solvng hat transfr problms and that s why som of th dfntons usd n ths concpt must b xpland [8]. Frst of all, th dfnton of a fuzzy st wll b ntroducd. Th fuzzy st n a nonmpty unvrsal st ( ) can b xprssd by a st of pars consstng of th lmnts and a crtan dgr of pr-assumd mmbrshp of th form x whr functon ( x) s dfnd as follows ( x) x, ( x) ; x () : 0, () In fuzzy sts, ach lmnt s mappd to [0,] by mmbrshp functon ( x), whr [0,] mans ral numbrs btwn 0 and (ncludng 0 and ). Consquntly, a fuzzy st s a vagu boundary st compard wth a crsp st. For vry x can b consdrd thr typs of mmbrshp to th fuzzy st :. ( x) full mmbrshp to th fuzzy st, x,. ( x) 0 lack of mmbrshp to th fuzzy st, x, 3. 0 ( x) partal mmbrshp to th fuzzy st. Th -cut st n unvrsal st than 0, [8,9] for vry s mad up of mmbrs whos mmbrshp s not lss x : ( x) (3) Th valu s arbtrary and ths -cut st s a crsp st. Ths st s dtrmnd by th followng charactrstc functon: for ( x) 0 for ( x) Evry fuzzy st can b dfnd as a sum of all ts -cuts (4) α (5) 0,

3 . Pascka-Blkhayat, P. Kowalsk whr α s a fuzzy st n th unvrs, whos mmbrshp functon s th followng for x 0 for x rthmtcal opratons ar gnrally vry complcatd. mong th nfnt quantty of possbl fuzzy sts that can b qualfd as fuzzy numbrs, som typs of mmbrshp functons ar of partcular mportanc. Du to ts rathr smpl mmbrshp functon of a lnar typ, trangular fuzzy numbrs ar on of th most frquntly usd fuzzy numbrs and n ths papr, trangular fuzzy numbrs wll b usd to solv th problm analysd. trangular fuzzy numbr a s a st wth th followng mmbrshp functon [0] whr a 0 0, x a x a, a x a0 a0 a a ( x) a x, a0 x a a a0 0, x a s th cor of th numbr, a, a ar th lft and th rght nd of th numbr rspctvly. trangular fuzzy numbr can b wrttn as a a, a0, a. On of th ways to avod vry complcatd arthmtc opratons prformd on fuzzy numbrs s to apply -cuts of fuzzy numbrs. In ths cas, th mathmatcal opratons ar dfnd accordng to th ruls of th drctd ntrval arthmtc prformd for vry -cut. Th a : 0, and -cut of a fuzzy numbr a : 0, s calld a st of closd ntrvals [8] whch satsfs th followng condtons a dfnd by a par of functons 0, a a, a (8). a :. a : 3. a a a a whr a ( a ) s a lmtd, monotonc functon for vry 0,. Evry fuzzy numbr can b prsntd as a sum of all ts own -cuts a [0, ] (6) (7) (9) a (0) 3

4 . Pascka-Blkhayat, P. Kowalsk Th -cut of a trangular fuzzy numbr a a, a0, a th form s th st of all closd ntrvals n 0, 0, 0 a a a a a a a Th dcomposton of a fuzzy numbr allows on to mak th mathmatcal opratons on closd ntrvals whch ar -cuts. In ths stuaton, complcatd arthmtc opratons can b omttd and t s possbl to apply th ntrval arthmtc for vry -cut. Th mathmatcal opratons ar smplfd, bcaus thy ar don only on th nds of th ntrvals. For th -cuts of two fuzzy numbrs b dfnd ( 0, ) [0]: addton subtracton multplcaton a and b,, () th followng mathmatcal opratons can a b a b a b (), a b a b a b (3) a b mn a b, a b, a b, a b, max a b, a b, a b, a b dvson ( 0 b, b ) (4) a a a a a a a a a mn,,,, max,,, b b b b b b b b b pplyng -cuts of th fuzzy numbrs allows on to us drctd ntrval arthmtc. (5) 3 FUZZY GOVERNING EQUTIONS Bo-hat transfr procdng n th htrognous skn tssu doman of th thcknss can b dscrbd by th systm of fuzzy nrgy quatons wrttn n th followng form [5,,,3] T( x, t) T( x, t) L x L : c Q (, ) x t t x whr =,, 3 corrsponds to th succssv layrs of skn such as pdrms, drms, hypodrms (s Fgur ), cs th fuzzy volumtrc spcfc hat, s th fuzzy thrmal conductvty, Q s th capacty of fuzzy ntrnal hat sourcs, T s th tmpratur, x and t dnot spatal co-ordnat and tm. Th capacty of fuzzy ntrnal hat sourcs s a sum of two componnts L 3 (6) 4

5 . Pascka-Blkhayat, P. Kowalsk Q( x, t) GB c B TB T ( x, t) Qm (7) whr G B s th fuzzy prfuson coffcnt, th artral blood tmpratur, Q m c B s th volumtrc spcfc hat of blood, s th fuzzy mtabolc hat sourc [4,5]. T B s Fgur : Skn tssu [6] Th fuzzy quatons (6) must b supplmntd by th boundary condtons and th ntal condton of th form whr q b T n T x L q x t n x 0: q( x, t) qb 3 3: (, ) (8) t 0: T ( x, 0) T (9) s th gvn fuzzy valu of th xtrnal hat sourc, T 0 s th ntal tmpratur. Btwn th succssv sub-domans th contnuty condton s takn nto account [5,,] T( x, t) T ( x, t) λ λ x L : n n T( x, t) T ( x, t) whr =,. Th quatons (6) (0) crat th mathmatcal modl of th procss dscussd. Th problm formulatd has bn solvd by mans of fuzzy fnt dffrnc mthod usng -cuts and th ruls of drctd ntrval arthmtc [4,5,0,7]. (0) 5

6 . Pascka-Blkhayat, P. Kowalsk 4 FUZZY FINITE DIFFERENCE METHOD Th problm analysd has bn solvd usng th fuzzy FDM wth α-cuts and th ruls of f f drctd ntrval arthmtc. t frst, th tm grd wth a constant stp t t t s ntroducd 0 f f f F t t... t t t... t () Th lft-hand sd of th fuzzy nrgy quatons (6) for th tm a dffrntal quotnt c f T T f f T ( x, t) f c t t t f can b substtutd by ddtonally, th frst trm of th rght-hand sd of th nrgy quatons can b transformd usng th dffrntal approxmaton of th scond drvatv whr x T T T x f f f f (, ) f x T x t s th msh stp and s th ndx of th cntral pont of star [5]. Fnally, on obtans th followng dffrntal ntrval quatons T b T b T T t f c f f f f f f GB cb TB T Qm f () (3) (4) whr b f t f c x (5) Usng th quaton (4) th tmpratur at th pont for tm lvl f can b found undr th assumpton that th stablty condton for xplct dffrntal schm s fulflld []. Th FDM approxmaton of th boundary condtons s constructd n a smlar way as n th papr [8]. Th boundary nods ar locatd at th dstanc 0.5 x wth rspct to th ral boundary. Ths approach gvs a bttr approxmaton of th Numann and Robn boundary condtons. Th boundary condton of th fourth typ s th condton of hat flow contnuty at th contact of two subdomans ( and + ). Th doman consdrd s covrd by a rgular gomtrcal msh (Fgur ). 6

7 . Pascka-Blkhayat, P. Kowalsk Fgur : Th rgular gomtrcal msh Th boundary condton of th fourth typ for dal contact can b wrttn usng th followng quatons T T x L f f T T f f λ λ : n n s mntond, th mathmatcal manpulatons ladng to th dsgnaton of tmpratur fld corrspondng to tm lvl f should b don usng α-cuts accordng to th ruls of drctd ntrval arthmtc [4,6,9]. 5 RESULTS OF COMPUTTIONS s a numrcal xampl th bo-hat transfr n a skn tssu of thcknss L3 =. mm has bn analysd. Th followng nput data hav bn ntroducd []: L = 0. mm, L =. mm, c = J/(m 3 K), G G = (m 3 blood/s)/m 3 tssu, B T B = 37C, G B = 0, 3 ntal tmpraturs T0 = T0 = T30 = 37 C, th xtrnal hat sourc B B (6) q b = W/m, th tm stp t 0.00 s, th msh stp x ( L L ) / n whr n 5, n 30 and n3 60. Th fuzzy trangular numbrs of thrmal conductvts ( 0.05,, 0.05 ) volumtrc spcfc hats c ( c 0.05 c, c, c 0.05 c ) and mtabolc hat sourcs Q ( Q 0.05 Q, Q, Q 0.05 ) hav bn ntroducd (for =,, 3) whr m m m m m = 0.35 W/(mK), c = J/(m 3 K), = W/(mK), c 3 3= 0.85 W/(mK), J/(m 3 K), = J/(m 3 K), Qm 0 W/m 3 and Q Q 45 W/m 3. Th tm of xtrnal hat sourc xposton has bn assumd as 5 s. m m3 Fgur 3 prsnts th hatng and coolng curvs at th slctd ntrnal nods L (), () for chosn valus of paramtr α. It should b pontd out that for ach nod of th doman consdrd thr ar two curvs rprsntng th bgnnng and nd of tmpratur ntrvals. Fgur 4 prsnts th ntrval valus of tmpraturs for th chosn paramtr α at th nod corrspondng to nod aftr 0 s. Ths fgur shows th dcomposton of th trangular fuzzy numbr bng th tmpratur valu and th dpndnc btwn th valu of th paramtr α and th wdth of th rsultng tmpratur ntrval. L c = L 7

8 . Pascka-Blkhayat, P. Kowalsk T[ o C] α = 0 T[ o C] α = t[s] t[s] T[ o C] α = 0.5 T[ o C] α = t[s] t[s] 0 Fgur 3: Th hatng and coolng curvs for chosn valus of paramtr α 0,75 0,5 0,5 0 39, 39,7 39,5 39,8 39,34 T[ o C] Fgur 4: Th ntrval tmpratur valus at th nod corrspondng to th nod L aftr 0 s for chosn valus of paramtr α 8

9 . Pascka-Blkhayat, P. Kowalsk 6 CONCLUSIONS In th papr, th numrcal analyss of a hat transfr procss procdng n th nonhomognous bologcal tssu doman has bn prsntd. In th mathmatcal modl, th thrmophyscal paramtrs of ach skn s layr (pdrms, drms, subcutanous rgon) such as volumtrc spcfc hat and thrmal conductvty hav bn gvn as fuzzy numbrs. Th problm analysd has bn solvd usng a fuzzy vrson of th fnt dffrnc mthod (FDM) wth -cuts and th ruls of drctd ntrval arthmtc. Such an approach allows on to avod complcatd fuzzy arthmtc and trat th consdrd fuzzy numbrs as ntrval numbrs. For bggr valus of, th tmpratur ntrval s narrowr. For th wdnss of th tmpratur ntrval s qual to 0. Th fuzzy vrson of FDM allows on to fnd th numrcal soluton n th fuzzy form and such nformaton may b mportant spcally for th paramtrs whch ar dffcult to stmat, for xampl tssu paramtrs. CKNOWLEDGMENTS, Th rsarch s fundd from th projcts Slsan Unvrsty of Tchnology, Faculty of Mchancal Engnrng BK-0/RMT-4/07 (0/040/BK 7/0045). REFERENCES [] Majchrzak, E. pplcaton of dffrnt varants of th BEM n numrcal modlng of bohat transfr problms. MCB: Mol. Cll. Bomch. (03), 0, 3: 0-3. [] Majchrzak, E., Mochnack, B., Jasńsk, M. Numrcal modllng of bohat transfr n mult-layr skn tssu doman subjctd to a flash fr, Computatonal Flud and Sold Mchancs (003), Volums and : [3] Cslsk, M. and Mochnack, B. pplcaton of th control volum mthod usng th Vorono polygons for numrcal modlng of bo-hat transfr procsss, J. Thor. ppl. Mch. (04), Volum 5, Issu 4: [4] Pascka-Blkhayat,. and Korczak,. Numrcal modllng of th transnt hat transport n a thn gold flm usng th fuzzy lattc Boltzmann mthod wth α-cuts. J. ppl. Math. Comput. Mch. (06), 5 (): [5] Mochnack, B. and Pascka-Blkhayat,. Numrcal modlng of skn tssu hatng usng th ntrval fnt dffrnc mthod. MCB: Mol. Cll. Bomch. (03), 0, 3: [6] Markov S.M. On drctd ntrval arthmtc and ts applcatons. J. Unvrs. Comput. Sc. (995), : [7] Numar. Intrval mthods for systm of quatons. Cambrdg Unvrsty Prss, Cambrdg, Nw York, Port Chstr, Mlbourn, Sydny (990). [8] Gachtt, R.E. and Young, R.E. paramtrc rprsntaton of fuzzy numbrs and thr arthmtc oprators. Fuzzy St Syst. (997), 9: [9] Gurra, M.L. and Stfann, L. pproxmat fuzzy arthmtc opratons usng monotonc ntrpolatons. Fuzzy St Syst. (005), 50: [0] Hanss, M. ppld Fuzzy rthmtc. Sprngr-Vrlag Brln Hdlbrg (005). [] Majchrzak, E. Bomchancs. Vol. XII, Polsh cadmy of Scncs, Warsaw (0). 9

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